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Properties

Label 2-315-63.16-c1-0-2
Degree $2$
Conductor $315$
Sign $-0.983 + 0.180i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.34 + 2.33i)2-s + (−1.34 − 1.09i)3-s + (−2.63 + 4.57i)4-s + 5-s + (0.739 − 4.61i)6-s + (−2.16 + 1.51i)7-s − 8.84·8-s + (0.613 + 2.93i)9-s + (1.34 + 2.33i)10-s − 3.92·11-s + (8.53 − 3.26i)12-s + (0.993 + 1.72i)13-s + (−6.46 − 3.01i)14-s + (−1.34 − 1.09i)15-s + (−6.64 − 11.5i)16-s + (2.23 + 3.87i)17-s + ⋯
L(s)  = 1  + (0.953 + 1.65i)2-s + (−0.776 − 0.630i)3-s + (−1.31 + 2.28i)4-s + 0.447·5-s + (0.301 − 1.88i)6-s + (−0.819 + 0.573i)7-s − 3.12·8-s + (0.204 + 0.978i)9-s + (0.426 + 0.738i)10-s − 1.18·11-s + (2.46 − 0.941i)12-s + (0.275 + 0.477i)13-s + (−1.72 − 0.806i)14-s + (−0.347 − 0.282i)15-s + (−1.66 − 2.87i)16-s + (0.542 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.983 + 0.180i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.983 + 0.180i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.109520 - 1.20067i\)
\(L(\frac12)\) \(\approx\) \(0.109520 - 1.20067i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.34 + 1.09i)T \)
5 \( 1 - T \)
7 \( 1 + (2.16 - 1.51i)T \)
good2 \( 1 + (-1.34 - 2.33i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + 3.92T + 11T^{2} \)
13 \( 1 + (-0.993 - 1.72i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.23 - 3.87i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0804 + 0.139i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.22T + 23T^{2} \)
29 \( 1 + (0.384 - 0.666i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.01 + 6.95i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.50 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.42 - 2.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.53 - 2.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.83 - 4.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.50 - 9.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.68 + 2.91i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.44 - 5.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.04 - 8.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.78T + 71T^{2} \)
73 \( 1 + (7.20 + 12.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.65 + 9.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.40 - 4.17i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.16 + 8.93i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.51 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.66852353430761741057386919144, −11.69314973617496980376256970405, −10.25102340962746443077677822699, −8.905319910444210519966283678491, −7.928383899781153998491256417017, −7.01476617244700152155229770666, −6.10842756265086651268846209707, −5.63793701010371917629863539473, −4.59292288530169783534232300844, −2.90763533271215708065468569873, 0.71128442649850960393989929561, 2.81253573045480082108514668187, 3.69477768896800348513284586949, 5.05369214302567849311113189659, 5.50423415359921707576182020252, 6.83756238346687869398138259432, 9.019354973750452252035211684583, 9.966490902255085577995153855930, 10.41641883715685561840812536702, 11.06083954856214648541138507145

Graph of the $Z$-function along the critical line